I guess they all can be broken back down into addition but I just have always had this burning question if there was some other mystery operator after exponentiation.
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http://en.wikipedia.org/wiki/Tetration – Dan Uznanski Mar 10 '15 at 01:19
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I believe so, but it's okay... Same question may bring up better answers as time progresses.... @DavidK – Panglossian Oporopolist Mar 10 '15 at 01:33
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What comes after tetration ? And after ? And after ? etc. also seems relevant. – David K Mar 10 '15 at 01:36
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@Kugelblitz I don't see any great harm in a duplicated question either, but there is still benefit marking a question duplicate because you then get all the previous answers at no additional charge. Even if the new question is closed due to duplication one can always post new answers to the original question. – David K Mar 10 '15 at 01:39
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@DavidK Agreed sir. I was under the impression that the OP was new, but it doesn't look like it, so it's fine like you say. – Panglossian Oporopolist Mar 10 '15 at 01:45
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Indeed. All of these come under the class of hyperoperators: http://en.wikipedia.org/wiki/Hyperoperation
So after Exponentiation, you have Tetration: http://en.wikipedia.org/wiki/Tetration.
Example: 
Then pentation: http://en.wikipedia.org/wiki/Pentation
For further study, you can read about this:
- http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
- http://en.wikipedia.org/wiki/Ackermann_function
- http://en.wikipedia.org/wiki/Conway_chained_arrow_notation
- http://en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation
Also, as suggested in the comments, you may want to take a look at this question, which is similar to yours... What comes after exponents?
Panglossian Oporopolist
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There is also repeated powering, e.g.$(...(((r^r)^r)^r)...)^r$, which is well defined for the positive reals. – Michael Ejercito Feb 01 '24 at 19:52
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There is the tower exponential $m^{m^{m \ldots}}$ where you do the tower $n$ times high in order to get the value for $m,n$. And if you want to keep going, at least for integers, you can look up the Ackerman function. That is sort of the well-known "ultimate" in defining a sequence of functions that all grow way faster than the previous one in the sequence, in a somewhat canonical fashion.
user2566092
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